This lesson is a cornerstone of geometry and a very common source of exam questions. It introduces the concepts of proportionality and similarity in triangles. Mastering the theorems in this lesson is key to solving complex geometry riders.

1. Key Definitions

• Equiangular Triangles: Two triangles are equiangular if the three angles of one triangle are equal to the three angles of the other triangle.

• Similar Triangles: For triangles, "similar" and "equiangular" mean the same thing. If two triangles are equiangular, they are similar. This means they have the same shape, but not necessarily the same size.

• Proportional Sides: The lengths of the corresponding sides of similar triangles have the same ratio.

2. The Core Theorems

This lesson is built on four main theorems (two main theorems and their converses).

Theorem 1: The Proportionality Theorem

"A line drawn parallel to a side of a triangle divides the other two sides proportionally."

• In simple terms: In ΔABC, if you draw a line PQ parallel to BC, then the ratio AP/PB will be exactly the same as the ratio AQ/QC.

• When to use it: When you are given parallel lines inside a triangle and need to find the length of a side segment.

Theorem 2: Converse of the Proportionality Theorem

"If a straight line divides two sides of a triangle proportionally, then that line is parallel to the remaining side."

• In simple terms: In ΔABC, if you can show that AP/PB = AQ/QC, then you can conclude that PQ must be parallel to BC.

• When to use it: When you need to prove that two lines are parallel.

Theorem 3: Equiangular Triangles Theorem

"The corresponding sides of two equiangular triangles are proportional."

• In simple terms: If you can prove that two triangles have the same three angles (ΔABC and ΔPQR), then you can state that their sides are in the same ratio: AB/PQ = BC/QR = AC/PR.

• When to use it: This is the most powerful theorem. Use it when you need to calculate unknown side lengths by proving two triangles are similar first.

Theorem 4: Converse of the Equiangular Triangles Theorem

"If the corresponding sides of two triangles are proportional, then those triangles are equiangular."

• In simple terms: If you can show that AB/PQ = BC/QR = AC/PR, then you can conclude that the triangles are equiangular (∠A=∠P, ∠B=∠Q, ∠C=∠R).

• When to use it: When you are given all the side lengths and need to prove that two triangles are similar (equiangular).

How to Identify Corresponding Sides (Crucial Skill!)

This is the most common place to make a mistake. The "corresponding sides" are the sides opposite the equal angles.

Example: If in ΔABC and ΔPQR, you know that ∠A = ∠P, ∠B = ∠Q, and ∠C = ∠R:

• Side BC is opposite ∠A. Side QR is opposite ∠P. So, BC corresponds to QR.

• Side AC is opposite ∠B. Side PR is opposite ∠Q. So, AC corresponds to PR.

• Side AB is opposite ∠C. Side PQ is opposite ∠R. So, AB corresponds to PQ.

The correct ratio is therefore: BC/QR = AC/PR = AB/PQ.

Exam Tips & Common Problems

• Tip 1: Prove Similarity First! Before you can use the proportionality of sides (Theorem 3), you must first prove that the two triangles are equiangular. A common exam structure is:

  • Part (i): Prove that ΔABC is equiangular to ΔPQR. (You need to find three pairs of equal angles using geometry rules like 'alternate angles', 'corresponding angles', 'vertically opposite angles', etc.)

  • Part (ii): Hence, find the length of side X. (Now you can use the proportionality theorem).

• Shortcut for Proving Similarity: You only need to prove that two pairs of angles are equal. Since the sum of angles in any triangle is 180°, the third pair must automatically be equal. You can write "(Remaining angles of the triangles)" as the reason.

• Common Problem: A small triangle inside a larger one, created by a parallel line (like Example 2 in the textbook).

  • How to solve: Prove that the small triangle (e.g., ΔAPQ) is equiangular to the large triangle (ΔABC).

    1. The angle at the top (∠A) is common to both triangles.

    2. The other two angles are equal because of corresponding angles (since PQ || BC).

    3. Once similarity is proven, you can set up the ratio of sides to find unknown lengths. Be careful: the sides of the small triangle correspond to the entire sides of the large triangle (e.g., AP corresponds to AB, not PB).